Diffusion-Controlled Evaporating Perfectly wetting meniscus

The Stefan diffusion theory is often used to predict the evaporation rate from capillaries. Following Derjaguin et al (Bull. Rilem, 1965), we show here that because the simple diffusion theory does not include mass loss from the precursor film, it significantly under-estimates mass loss from small ($\mu$m-sized) capillaries. We make the following simplifying assumptions. $(a)$ Diffusion is rate limiting; as a result, liquid and vapor at the interface are in local thermodynamic equilibrium. $(b)$ The system is effectively isothermal; though evaporation induces liquid temperature differences, they are kinetically negligible. $(c)$ Transport of the evaporating molecules is by axial-diffusion only. At a given axial location $x$, the total mass flow $\dot{m}$ is the sum of the flow within the thin liquid film and that occurring by diffusion within the gas $-\dot{m} = \frac{h^3}{3\nu}\frac{dp_\ell}{dx} + \frac{\rho_sD} {\rho RT_w}(a-h)\frac{dp_\ell}{dx} = constant$ where $\nu$ is the kinematic viscosity, $R$ the gas constant, $D$ the diffusion coefficient, $\rho$ the liquid density, $T_w$ the wall temperature, $2a$ the channel gap thickness, and $\rho_s$ the saturation density. The interface shape $h(x)$ is related to the liquid pressure at the interface $p_\ell(x)$ by the normal stress equation $P_a - p_\ell = \gamma\frac{d^2h}{dx^2} + \frac{A}{h^3}$ where $P_a$ is the total gas pressure and $A$ the Hammaker constant. Our model differs from that of Derjaguin et al because including surface tension $\gamma$ allows us to calculate the shape of the whole meniscus, and so to relate our results for mass loss to that given by standard theory.